Exercise 4 (7 points) Themes: numerical functions, exponential function
Part A: study of two functions Consider the two functions $f$ and $g$ defined on the interval $[0; +\infty[$ by: $$f(x) = 0.06\left(-x^2 + 13.7x\right) \quad \text{and} \quad g(x) = (-0.15x + 2.2)\mathrm{e}^{0.2x} - 2.2.$$ We admit that the functions $f$ and $g$ are differentiable and we denote $f'$ and $g'$ their respective derivative functions.
- The complete table of variations of function $f$ on the interval $[0; +\infty[$ is given.
| $x$ | 0 | 6.85 | $+\infty$ |
| \multirow{2}{*}{$f(x)$} | | $\nearrow f(6.85)$ | |
| 0 | | $\underline{-}_{\infty}$ |
a. Justify the limit of $f$ at $+\infty$. b. Justify the variations of function $f$. c. Solve the equation $f(x) = 0$. - a. Determine the limit of $g$ at $+\infty$. b. Prove that, for every real $x$ belonging to $[0; +\infty[$ we have: $g'(x) = (-0.03x + 0.29)\mathrm{e}^{0.2x}$. c. Study the variations of function $g$ and draw its table of variations on $[0; +\infty[$. Specify an approximate value to $10^{-2}$ of the maximum of $g$. d. Show that the equation $g(x) = 0$ has a unique non-zero solution and determine, to $10^{-2}$ near, an approximate value of this solution.
Part B: trajectories of a golf ball We wish to use the functions $f$ and $g$ studied in Part A to model in two different ways the trajectory of a golf ball. We assume that the terrain is perfectly flat. We will admit here that 13.7 is the value that cancels the function $f$ and an approximation of the value that cancels the function $g$. For $x$ representing the horizontal distance traveled by the ball in tens of yards after the shot (with $0 < x < 13.7$), $f(x)$ (or $g(x)$ depending on the model) represents the corresponding height of the ball above the ground, in tens of yards. The ``takeoff angle'' of the ball is called the angle between the x-axis and the tangent to the curve ($\mathscr{C}_f$ or $\mathscr{C}_g$ depending on the model) at its point with abscissa 0. A measure of the takeoff angle of the ball is a real number $d$ such that $\tan(d)$ is equal to the slope of this tangent. Similarly, the ``landing angle'' of the ball is called the angle between the x-axis and the tangent to the curve ($\mathscr{C}_f$ or $\mathscr{C}_g$ depending on the model) at its point with abscissa 13.7. A measure of the landing angle of the ball is a real number $a$ such that $\tan(a)$ is equal to the opposite of the slope of this tangent. All angles are measured in degrees.
- First model: recall that here, the unit being tens of yards, $x$ represents the horizontal distance traveled by the ball after the shot and $f(x)$ the corresponding height of the ball. According to this model: a. What is the maximum height, in yards, reached by the ball during its trajectory? b. Verify that $f'(0) = 0.822$. c. Give a measure in degrees of the takeoff angle of the ball, rounded to the nearest tenth. (You may use the table below). d. What graphical property of the curve $\mathscr{C}_f$ allows us to justify that the takeoff and landing angles of the ball are equal?
- Second model: recall that here, the unit being tens of yards, $x$ represents the horizontal distance traveled by the ball after the shot and $g(x)$ the corresponding height of the ball. According to this model: a. What is the maximum height, in yards, reached by the ball during its trajectory? We specify that $g'(0) = 0.29$ and $g'(13.7) \approx -1.87$. b. Give a measure in degrees of the takeoff angle of the ball, rounded to the nearest tenth. (You may use the table below). c. Justify that 62 is an approximate value, rounded to the nearest unit, of a measure in degrees of the landing angle of the ball.
Table: excerpt from a spreadsheet giving a measure in degrees of an angle when its tangent is known:
| A | B | C | D | E | F | G | H | I | J | K | L | M |
| 1 | $\tan(\theta)$ | 0.815 | 0.816 | 0.817 | 0.818 | 0.819 | 0.82 | 0.821 | 0.822 | 0.823 | 0.824 | 0.825 | 0.826 |
| 2 | $\theta$ in degrees | 39.18 | 39.21 | 39.25 | 39.28 | 39.32 | 39.35 | 39.39 | 39.42 | 39.45 | 39.49 | 39.52 | 39.56 |
| 3 | | | | | | | | | | | | | |
| 4 | $\tan(\theta)$ | 0.285 | 0.286 | 0.287 | 0.288 | 0.289 | 0.29 | 0.291 | 0.292 | 0.293 | 0.294 | 0.295 | 0.296 |
| 5 | $\theta$ in degrees | 15.91 | 15.96 | 16.01 | 16.07 | 16.12 | 16.17 | 16.23 | 16.28 | 16.33 | 16.38 | 16.44 | 16.49 |
Part C: Interrogating the models Based on a large number of observations of professional players' performances, the following average results were obtained:
| Launch angle in degrees | Maximum height in yards | Landing angle in degrees | Horizontal distance in yards at the point of impact |
| 24 | 32 | 52 | 137 |
Which model, among the two previously studied, seems most suitable for describing the ball strike by a professional player? The answer will be justified.